Generalization of the Kermack-McKendrick SIR Model to a Patchy Environment for a Disease with Latency

J. Li¹ ; X. Zou¹

¹ Department of Applied Mathematics University of Western Ontario London, Ontario, Canada N6A 5B7

Mathematical modelling of natural phenomena, Tome 4 (2009) no. 2, pp. 92-118.

Résumé

In this paper, with the assumptions that an infectious disease has a fixed latent period in a population and the latent individuals of the population may disperse, we reformulate an SIR model for the population living in two patches (cities, towns, or countries etc.), which is a generalization of the classic Kermack-McKendrick SIR model. The model is given by a system of delay differential equations with a fixed delay accounting for the latency and non-local terms caused by the mobility of the individuals during the latent period. We analytically show that the model preserves some properties that the classic Kermack-McKendrick SIR model possesses: the disease always dies out, leaving a certain portion of the susceptible population untouched (called final sizes). Although we can not determine the two final sizes, we are able to show that the ratio of the final sizes in the two patches is totally determined by the ratio of the dispersion rates of the susceptible individuals between the two patches. We also explore numerically the patterns by which the disease dies out, and find that the new model may have very rich patterns for the disease to die out. In particular, it allows multiple outbreaks of the disease before it goes to extinction, strongly contrasting to the classic Kermack-McKendrick SIR model.

Détail

DOI : 10.1051/mmnp/20094205

Bibliographie

[1] R. M. Anderson, R. M. May. Infectious diseases of humans: dynamics and control, Oxford University Press, Oxford, UK, 1991.

[2] J. Arino, P. van den Driessche. A multi-city epidemic model. Math. Popul. Stud., 10 (2003), 175-193.

[3] J. Arino, P. van den Driessche. The basic reproduction number in a multi-city compartmental epidemic model. LNCIS, 294 (2003), 135-142.

[4] F. Brauer. Some simple epidemic models. Math. Biosci. Engin., 3 (2006), 1-15.

[5] O. Diekmann, J. A. P. Heesterbeek. Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. Wiley, 2000.

[6] J. K. Hale, S. M. Verduyn Lunel. Introduction to functional differential equations. Spring-Verlag, New York, 1993.

[7] W. O. Kermack, A. G. McKendrick. A contribution to the mathematical theory of epidemics. Proc. Royal Soc. London, 115 (1927), 700-721.

[8] Y.-H. Hsieh, P. van den Driessche, L. Wang. Impact of travel between patches for spatial spread of disease. Bull. Math. Biol., 69 (2007), 1355-1375.

[9] J. A. J. Metz, O. Diekmann. The dynamics of physiologically structured populations. Springer-Verlag, New York, 1986.

[10] K. Mischaikow, H. Smith, H. R. Thieme. Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions. Trans. Amer. Math. Soc., 347 (1995), 1669-1685.

[11] J. D. Murray. Mathematical biology. 3rd ed., Springer-Verlag, New York, 2002.

[12] M. Salmani, P. van den Driessche. A model for disease transmission in a patchy environment. Disc. Cont. Dynam. Syst. Ser. B, 6 (2006), 185-202.

[13] H. R. Thieme, C. Castillo-Chavez. Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity, Vol. 1: Theory of Epidemics (O. Arino, D. Axelrod, M. Kimmel, M. Langlais eds.), pp. 33-50, Wuerz, 1995.

[14] W. Wang, X.-Q. Zhao. An epidemic model in a patchy environment. Math. Biosci., 190 (2004), 97-112.

[15] W. Wang, X.-Q. Zhao. An age-structured epidemic model in a patchy environment. SIAM J. Appl. Math., 65 (2005), 1597-1614.

[16] W. Wang, X.-Q. Zhao. An epidemic model with population dispersal and infection period. SIAM J. Appl. Math., 66 (2006), 1454-1472.